Definition 2.2 Let \(P \in (X\backslash D_{\textrm {red}})(F)\) . For a place \(v \notin S\) , let \({P}_v\)  denote the induced point in \({X}({O}_v)\)  obtained via the valuative criterion for properness as stated in [1]}. For \(\alpha \in {A}\) , we define the intersection multiplicity \(n_v({D}_{\alpha },P)\)  of \({D}_{\alpha }\)  and \(P\)  at \(v\)  to be the colength of the ideal \({P}_v^{*}{D}_{\alpha }\)  in \({O}_v\) . Then the intersection number of \(P\)  and \({D}\)  at \(v\)  is defined to be
\(n_v({D},P) = \sum _{\alpha \in {A}}\epsilon _{\alpha }n_v({D}_{\alpha },P).\) 

